Problem: Lucas programs his quadcopter to fly with a velocity (speed and direction) vector $\vec{p} = 4\hat i + 7\hat j$, which should send it straight at its target (in the absence of wind). To his dismay, his quadcopter actually moves with a velocity vector $\vec{a} = 2\hat i + 8\hat j$. (Speeds are given in meters per second, $\text{m}/\text{s}$.) What is the speed of the wind?
Explanation: Consider vector $\vec w$ (depicted below), which represents the wind. We can imagine how it would cause Lucas' quadcopter to change direction and possibly speed up or slow down a tiny bit. It is reasonable to assume that the velocity of the wind added with the programmed velocity of Lucas' quadcopter equals the resultant (actual) velocity of his quadcopter. $\vec w + \vec{p} = \vec{a}$ We can now solve for $\vec w$. $\begin{aligned} \vec w + \vec p &= \vec a\\\\ \vec w &= \vec a - \vec p\\\\ \vec w &= (2\hat i + 8\hat j) - (4\hat i + 7\hat j)\\\\ \vec w &= -2\hat i + 1\hat j \end{aligned}$ We can find the magnitude of $\vec w$ (i.e., the speed of the wind) using the Pythagorean theorem. $\begin{aligned} \| \vec w \|^2 &= -2^2 + 1^2\\\\ \| \vec w \| &= \sqrt{-2^2 + 1^2}\\\\ \| \vec w \| &= \sqrt{5}\\\\ \| \vec w \| &\approx 2.2 \text{ m}/\text{s} \end{aligned}$ $\vec w$ is pointing into the second quadrant, so we can find its direction (call it $\theta~$ ) using the arctangent function and adding $180^\circ$. $\begin{aligned} \tan \theta &= \dfrac{1}{-2}\\\\ \theta &= \arctan{ \left ( \dfrac{1}{-2} \right ) } \\\\ \theta&\approx -26.565^\circ \end{aligned}$ Adding $180^\circ$ to this result gives us the actual direction, $153^\circ$ (rounded to the nearest degree). The speed of the wind is $2.2 \text{ m}/\text{s}$. The direction of the wind is $153^\circ$.